C₅×#8544 #11931264

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#8544

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂⁵	C₂³×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

1222 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(6,7)(8,12,9,11)(10,13)(14,15)(16,22,24,19)(17,20)(18,21)(23,28,29,26)(25,27);(6,8,10,9)(7,11,13,12)(14,16)(15,19)(17,23)(18,24)(20,26)(21,22)(25,29)(27,28);(6,9,10,8)(7,12,13,11)(14,17,18,25)(15,20,21,27)(16,23,24,29)(19,26,22,28)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1532);